Taming the flow of light via active magneto-optical impurities

Hamidreza Ramezani; Zin Lin; Samuel Kalish; Tsampikos Kottos; Vassilios Kovanis; Ilya Vitebskiy

doi:10.1364/OE.20.026200

1. Introduction

The present rapid development of global communications and computer science demands constantly increasing capacities of data transmission and data processing. In this endeavor, photonic integrated circuits (PICs) are becoming very promising for many applications, including next generation optical networks, optical interconnects, tunable photonic oscillators, wavelength division multiplexed systems, coherent transceivers and optical buffers [1–3]. This is because PICs provide advantages, e.g., higher performance, reduced device footprint, lower component-to-component coupling losses and lower power consumption, to name a few. However, the progress on PICs is still quite limited. One challenging task that researchers are currently facing is the realization of novel classes of non-reciprocal integrated photonic devices that allow one-directional flow of information-e.g. optical isolators and circulators [4]. Currently, such unidirectional elements rely almost exclusively on magneto-optical effects, like the magnetic Faraday rotation caused by non-reciprocal circular birefringence.

At optical frequencies, all non-reciprocal effects (NRE), such as magnetic Faraday rotation, are very weak. This weakness of NRE results in prohibitively large size of most non-reciprocal devices. A natural way to enhance a weak NRE, and thus reduce the size of a structure, is to incorporate the magneto-optical material into an optical resonator, which can be a photonic structure with feature sizes comparable to the light wavelength. At the same time, such resonators cause undesirable effects, like enhanced absorption, linear birefringence, and nonlinear effects. The most deleterious of all is absorption which under resonance conditions can dramatically affect the functionality of the optical devices. For example, an enhanced absorption causes deviation of the transmitted light polarization from linear to elliptic (non-reciprocal circular dichroism), which significantly compromises the performance of optical isolator. In addition, the enhanced absorption can results in a significant power loss. Finally, even moderate absorption can lower the quality factor of the optical resonator by several orders of magnitude and, thereby, significantly compromise its performance as a Faraday rotation enhancer [5].

In this Letter we show that a strongly asymmetric transport at infrared and optical frequencies can be achieved in active magneto-photonic structures in which the spatial destribution of gain and loss displays a special, anti-linear symmetry (see Fig. 1). In classical optics, it involves the combination of delicately balanced gain and loss regions together with the modulation of the index of refraction [7]. A sub-class of such optical antilinear structures are the so–called parity-time (𝒫𝒯) symmetric media for which the complex index of refraction obeys the condition n(r⃗) = n^*(−r⃗). Synthetic materials with 𝒫𝒯 symmetries are shown to exhibit several intriguing features some of which have been already demonstrated in a series of recent experimental papers [9–13]. These include among others, power oscillations [7–10], absorption enhanced transmission [11], unidirectional transparency and invisibility [12, 14], non-reciprocal Bloch oscillations [15, 16], a new type of conical diffraction [17] and reconfigurable Talbot effects [18]. In the nonlinear domain, such asymmetric transmittance effects can be used to realize a new generation of optical on-chip isolators and circulators [19]. Other results include the realization of coherent perfect laser-absorber [20–22] and nonlinear switching structures [23].

Fig. 1 (a) Scattering setup of a micro-cavity with 𝒫̃𝒯 - symmetric gain/loss distribution and asymmetric transport. The left/right (green/red) slab is a lossy/gain non magnetic layer with in-plane anisotropy. The middle (arsenic) slab is a passive ferromagnetic material with magnetization M₀ (indicated with the arrows inside the layer). (b) A generalized 𝒫̃𝒯 - symmetric micro-cavity embedded in an anisotropic Bragg reflector.

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The micro-cavity that we investigate (see Fig. 1(a)) is a generalization of the standard 𝒫𝒯 -symmetric structures and consists of three components: a central magnetic layer sandwiched between two active (one with gain and the other with loss) anisotropic layers distributed in a way that the whole structure exhibits an antilinear symmetry. The magnetic layer provides a nonreciprocal circular birefringence (magnetic Faraday effect). The role of the magnetic layer is to break the Lorentz reciprocity, which would otherwise impose the symmetry in forward and backward transmission of the layered structure. The magnetic circular birefringence does break the Lorents reciprocity, but it is not sufficient to provide asymmetry in forward and backward transmission. Another requirement is a broken space inversion symmetry. This is achieved with the use of misaligned birefringent layers as suggested in [24]. Even then, the asymmetry in forward and backward transmission averaged over the input light polarization remains symmetric. In this report, we show that the interplay of the active birefringent elements with the magneto-optical layer can lead to strong asymmetry in forward and backward transmission. Although in such micro-cavities the local unitarity is violated, the balanced gain/loss design that we incorporated in our structure imposes generalized unitary relations for the energy flux conservation. The transport asymmetry can be drastically enhanced by embedding our micro-cavity inside an anisotropic Bragg grating (see Fig. 1(b)). We show the formation of broad frequency domains at the pseudo-gaps of the grating which support high-Q micro-cavity modes with enhanced transport asymmetry. The proposed architecture can be used for the creation of highly efficient on-chip non-reciprocal devices such as optical isolators and circulators.

2. Modeling and symmetries

We consider the structure shown in Fig. 1(a). The non-Hermitian permittivity tensor, ε̂(z) is assumed to be

\hat{ε} (z^{+}) = [\begin{matrix} ε_{x x} & ε_{x y} & 0 \\ ε_{x y}^{*} & ε_{y y} & 0 \\ 0 & 0 & ε_{z z} \end{matrix}], \hat{ε} (z^{-}) = [\begin{matrix} ε_{x x}^{*} & ε_{x y} & 0 \\ ε_{x y}^{*} & ε_{y y}^{*} & 0 \\ 0 & 0 & ε_{z z} \end{matrix}],

where the variables z^± take values in the intervals −L ≤ z − ≤ 0 and 0 ≤ z⁺ ≤ L. Above, ε_xx = ε(z)+δ(z)cos(2ϕ(z))+iγ(z), ε_xy = δ(z)sin(2ϕ(z))+iα(z) and ε_yy = ε(z)−δ(z)cos(2ϕ(z))+ iγ(z). The function δ describes the magnitude of in-plane anisotropy, γ is the gain/loss parameter and the angle ϕ defines the orientation of the principle axes in the xy–plane. The gyrotropic parameter α is responsible for the Faraday rotation. Outside the scattering region z ∉ [−L,L], we assume that the permittivity takes a constant value ε₀ i.e ε̂ = ε₀ × 1̂ where 1̂ is the 2 × 2 identity matrix.

The corresponding permeability tensor, μ̂(z) takes the form

\hat{μ} (- L \leq z \leq L) = [\begin{matrix} μ_{x x} & μ_{x y} & 0 \\ μ_{x y}^{*} & μ_{x x} & 0 \\ 0 & 0 & μ_{z z} \end{matrix}]; \hat{μ} (z \notin [- L, L]) = \hat{1}

where μ_xx = μ(z) = μ(−z), μ_xy = iβ(z) = iβ(−z) and β is another gyrotropic parameter which essentially depends on the static components of the magnetic field H⃗₀, as well as the frequency ω.

Changing H⃗₀ → −H⃗ ₀ and M⃗₀ → −M⃗₀ implies the following transformation [24]

\hat{ε} \to \hat{ε}, \hat{μ} \to \hat{μ}, α \to - α, β \to - β

which allows us to conclude that the micro-cavity of Fig. 1(a) is invariant under the combined 𝒫̃𝒯 antilinear symmetry:

\begin{array}{l} \tilde{𝒫} 𝒯 \hat{ε} (z) {(\tilde{𝒫} 𝒯)}^{- 1} = \hat{ε} (z) \\ \tilde{𝒫} 𝒯 \hat{μ} (z) {(\tilde{𝒫} 𝒯)}^{- 1} = \hat{μ} (z) \end{array}

The time reversal operator 𝒯 is an anti-linear operator which performs transpose complex conjugation while the linear operator 𝒫̃ = 𝒫Θ consist of the parity operator 𝒫 which represents a spatial inversion r⃗ → −r⃗ (note though that in our case spacial inversion is the same as reflection), and the exchange operator Θ which changes ϕ₊ ≡ ϕ(z⁺) ↔ ϕ₋ ≡ ϕ(z⁻). We note that in the case of ϕ₊ = ϕ₋, i.e. the two layers are not misaligned, the structure of Fig. 1(a) is 𝒫𝒯 -symmetric. For this reason, below we will refer to our structure (with misalignment), as a generalized 𝒫𝒯 -symmetric geometry.

3. Scattering formalism

The electric and magnetic field are designated by the time-harmonic Maxwell equations:

\nabla \times \vec{E} (\vec{r}) = i \frac{ω}{c} \hat{μ} \vec{H} (\vec{r}), \nabla \times \vec{H} (\vec{r}) = - i \frac{ω}{c} \hat{ε} \vec{E} (\vec{r})

with the following solution

\vec{E} (\vec{r}) = e^{i (k_{x} x + k_{y} y)} \vec{E} (z), \vec{H} (\vec{r}) = e^{i (k_{x} x + k_{y} y)} \vec{H} (z) .

Assuming normal propagation, i.e. k_x = k_y = 0, the solutions of Eq. (6) for E⃗(z) in the left (l) and right (r) side of the scattering region, are written in terms of the forward and backward traveling waves:

{\vec{E}}^{l, r} (α, β, ϕ, z) = A^{l, r} e^{i k z} + B^{l, r} e^{- i k z} .

where

\begin{array}{l} A^{l, r} = {[A_{x}^{l, r} (α, β, ϕ_{-}, ϕ_{+}) A_{y}^{l, r} (α, β, ϕ_{-}, ϕ_{+})]}^{T} \\ B^{l, r} = {[B_{x}^{l, r} (α, β, ϕ_{-}, ϕ_{+}) B_{y}^{l, r} (α, β, ϕ_{-}, ϕ_{+})]}^{T} \end{array}

The corresponding magnetic field H⃗(z) is

\vec{H} (z) = \frac{1}{c} \hat{z} \times \vec{E} (z)

, where ẑ is the unit vector in the z-direction.

The 4 × 4 transfer matrix M ≡ M(α, β, ϕ₋, ϕ₊) directly furnishes the relation between the electric field on the left and right sides of the scattering region (below we assume c = 1 units), i.e.,

[\begin{matrix} A^{r} \\ B^{r} \end{matrix}] = M [\begin{matrix} A^{l} \\ B^{l} \end{matrix}], M = [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] .

Performing the operation 𝒫̃𝒯 on the solutions of Eq. (7) results in

{\vec{E}}^{l, r} (α, β, ϕ_{-}, ϕ_{+}) \overset{\tilde{𝒫} 𝒯}{} {({\vec{E}}^{l, r} (- α, - β, ϕ_{+}, ϕ_{-}))}^{*}

Since our system is invariant under the 𝒫̃𝒯-operation then Eq. (10) is also a solution of the Maxwell equations. Applying once more the transfer matrix M(α, β, ϕ₋, ϕ₊) for the transformed solutions Eq.(10), we get:

{[\begin{array}{l} A^{l} (- α, - β, ϕ_{+}, ϕ_{-}) \\ B^{l} (- α, - β, ϕ_{+}, ϕ_{-}) \end{array}]}^{*} = M {[\begin{array}{l} A^{r} (- α, - β, ϕ_{+}, ϕ_{-}) \\ B^{r} (- α, - β, ϕ_{+}, ϕ_{-}) \end{array}]}^{*} .

It follows from the Eq.(11) and the conjugated form of the Eq.(9) with α → −α, β → −β, ϕ₋ ↔ ϕ₊ that

M (α, β, ϕ_{-}, ϕ_{+}) M^{*} (- α, - β, ϕ_{+}, ϕ_{-}) = \hat{1} .

The transmission and reflection coefficients can be expressed in terms of the transfer matrix elements as

{\begin{array}{l} r^{l} = - M_{22}^{- 1} M_{21}, & r^{r} = M_{12} M_{22}^{- 1} \\ t^{l} = M_{11} - M_{12} M_{22}^{- 1} M_{21}, & t^{r} = M_{22}^{- 1} \end{array}

which leads us to the conclusion that the transmission/reflection for left or right incidence are not necessarily the same.

From Eqs. (13) one can deduce the form of the scattering matrix $S \equiv S (α, β, ϕ_{-}, ϕ_{+}) = [\begin{array}{l} r^{l} & t^{r} \\ t^{l} & r^{r} \end{array}]$ , in terms of the M-matrix elements. After some straightforward algebra we can show that the scattering matrix should satisfy the following generalized unitary relation:

𝒫 S * (- α, - β, ϕ_{+}, ϕ_{-}) 𝒫 S (α, β, ϕ_{-}, ϕ_{+}) = \hat{1}

where

𝒫 = [\begin{array}{l} 0 & \hat{1} \\ \hat{1} & 0 \end{array}]

is the representation of the parity operator in the channel space. This is a fundamental relation obeyed by 𝒫̃𝒯 -symmetric S matrices in the presence of magneto-optical effects, and it is a weaker constraint than unitarity. In the latter case, each eigenvalue of the S-matrix is unimodular. In contrast, for generalized 𝒫̃𝒯 -symmetric structures this is not necessarily the case. Specifically, one can distinguish between two scenarios: If the S and 𝒫̃𝒯 operator share the same eigenvectors |e_n〉 with the corresponding S-eigenvalues e_n(α, β, ϕ₋, ϕ₊) then

e_{n}^{*} (- α, - β, ϕ_{+}, ϕ_{-}) e_{n} (α, β, ϕ_{-}, ϕ_{+}) = 1

, and the corresponding eigenstates |e_n〉 exhibit no net amplification or dissipation. In the opposite case |e_n〉 is not itself 𝒫̃𝒯- symmetric, but a pair satisfies 𝒫̃𝒯 by transforming into each other |e_m〉 ≡ 𝒫̃𝒯|e_n〉 ≠ |e_n〉 with one exhibiting amplification, and the other dissipation. The corresponding S-matrix eigenvalues satisfy the relation

e_{m}^{*} (- α, - β, ϕ_{+}, ϕ_{-}) e_{n} (α, β, ϕ_{-}, ϕ_{+}) = 1

.

4. Asymmetric transport

Next we focus on the transport properties of the simple 𝒫̃𝒯 -symmetric magneto-optical cavity of Fig. 1(a). The misalignment angle Δϕ = ϕ₊ −ϕ₋ between the pair of active layers is different from 0 and π/2 to ensure the asymmetry of forward and backward wave propagation [24]. In the absence of gain and loss, a polarization averaged asymmetric transmission would be prohibited due to the sum rule resulting from the flux conservation and unitarity of the scattering matrix [25]. This prohibition is lifted with the introduction of loss and gain layers in Fig. 1(a).

To confirm the above expectations we present in Fig. 2(a) the result of our numerical simulation for the average (left/right) reflectance <R^l,r>_p while in Fig. 2(b) we report the left-right transmittance difference |<T^l>_p − <T^r >_p| (where R ≡ |r|² and T = |t|²). In all these simulations the average 〈·〉_p is taken over all possible polarizations of the incoming wave. We find that the reflectances and transmittances for left/right incident waves are different from one another. Although an asymmetric left-right reflection is a characteristic property of systems with anti-linear symmetry [14], the asymmetry in the transmittances is a new element which essentially requires the presence of both a magneto-optical material and loss and/or gain materials. To further quantify the asymmetric behavior of our structure we report in Fig. 2(c) the asymmetry quality factor Q_T which is defined as

Q_{T} = \frac{| < T^{l} - T^{r} >_{p} |}{< T^{l} + T^{r} >_{p}} .

Fig. 2 (a) The left/right reflectances vs. ω. (b) The difference |〈T_l〉 − 〈T_r〉| between the left and right transmittances vs. ω. (c) the quality factor Q_T vs. ω. All layers have the same thickness d, the phase misalignment is $Δ ϕ_{active dielectric} = \sqrt{π}$ , the anisotropy is δ = 1, while the gain/loss parameter is γ = 0.0005. Furthermore the real part of permittivity for the birefringent layers is ε = 9, for the magneto-optical layer is ε = 1.525 while we assume that ε₀ = 1. The Faraday gyrotropic parameters are α = 0.925, β = 0, and μ = 1.

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The asymmetry of the 𝒫̃𝒯 -symmetric magneto-optical micro-cavity can be further amplified by embedding it between two identical anisotropic Bragg mirrors, see Fig. 1(b). The anisotropy at the Bragg gratings creates pseudo-gaps at the transmission spectrum as shown in Fig. 3(a). We have found that at these frequency windows the non-reciprocity is enhanced. In Fig. 3(b) we report the Q_T -factor for a structure shown in Fig. 1(b) with a grating consisting of only 45 layers. The frequency domains of polarization independent asymmetric transport are marked with (green) shadowed areas and coincide with the pseudo-gaps of the grating.

Fig. 3 (Upper subfigure) Dispersion relation ω(k) of the infinite periodic anisotropic Bragg reflector with permittivity contrasts ε = 3 and ε = 12 and δ = 1, Δϕ = 0. (Lower subfigure) Quality factor Q_T (red dashed line) of a 𝒫̃𝒯 -symmetric non-reciprocal magneto-optical cavity when it is embedded in the anisotropic Bragg reflector associated with the set-up of the upper sub-figure. The micro-cavity has the same constitute parameters as in Fig. 2. The transmittance spectrum of the periodic Bragg is also shown in order to identify the pseudo-gap frequency windows (green shadowed areas) where the Q_T -factor take large values.

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The enhancement of asymmetry can be understood intuitively once we realize that our micro-cavity behaves as a high-Q optical resonator filled with magneto-optical material once it is embedded in a Bragg grating. In this case, each individual photon resides in the magnetic material much longer compared to the same piece of magnetic material placed outside the resonator. Since the sign of Faraday rotation is independent of the direction of light propagation, one can assume that the total amount of Faraday rotation is proportional to the photon residence time in the magnetic material.

5. Conclusion

Although the enhancement of non-reciprocal effects, such as Faraday rotation, can be achieved via resonance conditions (see, for example [26–28] and references therein), the same resonance conditions also enhance the absorption, which can ruin the performance of almost any non-reciprocal device. Here we have shown that the use of a judiciously balanced gain and loss unit shown in Fig. 1, or its equivalent, can simultaneously solve two fundamental problems. Firstly, it allows a significant enhancement of the desired non-reciprocal effects without creating the energy loss issue. Secondly, the simultaneous presence of loss/gain components and a magnetic component result in strong polarization-independent transmission asymmetry, which is the defining property of an optical isolator. Our photonic architecture may be used as the building block for chip-scale non-reciprocal devices such as optical isolators and circulators.

Acknowledgments

This research was supported by an AFOSR No. FA 9550-10-1-0433 grant and LRIR 09RY04COR grant, and by an NSF ECCS-1128571 grant.

References and links

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Taming the flow of light via active magneto-optical impurities

Abstract

1. Introduction

2. Modeling and symmetries

3. Scattering formalism

4. Asymmetric transport

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (15)

Optics Express